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2021 Semigroups for one-dimensional Schrödinger operators with multiplicative Gaussian noise
Pierre Yves Gaudreau Lamarre
Author Affiliations +
Electron. J. Probab. 26: 1-47 (2021). DOI: 10.1214/21-EJP654


Let H:=12Δ+V be a one-dimensional continuum Schrödinger operator. Consider Hˆ:=H+ξ, where ξ is a translation invariant Gaussian noise. Under some assumptions on ξ, we prove that if V is locally integrable, bounded below, and grows faster than log at infinity, then the semigroup etHˆ is trace class and admits a probabilistic representation via a Feynman-Kac formula. Our result applies to operators acting on the whole line R, the half line (0,), or a bounded interval (0,b), with a variety of boundary conditions. Our method of proof consists of a comprehensive generalization of techniques recently developed in the random matrix theory literature to tackle this problem in the special case where Hˆ is the stochastic Airy operator.

Funding Statement

This research was partially supported by an NSERC doctoral fellowship and a Gordon Y. S. Wu fellowship.


The author thanks Laure Dumaz for an insightful discussion on the one-dimensional Anderson Hamiltonian and the parabolic Anderson model, which served as a chief motivation for the writing of this paper. The author thanks Michael Aizenman for helpful pointers in the literature regarding random Schrödinger operators. The author gratefully acknowledges Mykhaylo Shkolnikov for his continuous guidance and support and for his help regarding a few technical obstacles in the proofs of this paper, as well as Vadim Gorin and Mykhaylo Shkolnikov for discussions concerning the resolution of an error that appeared in a previous version of the paper.

The author thanks anonymous referees for carefully reading several previous versions of this paper, as well as a number of insightful comments that helped significantly improve the presentation of the present version.


Download Citation

Pierre Yves Gaudreau Lamarre. "Semigroups for one-dimensional Schrödinger operators with multiplicative Gaussian noise." Electron. J. Probab. 26 1 - 47, 2021.


Received: 31 October 2019; Accepted: 7 June 2021; Published: 2021
First available in Project Euclid: 20 July 2021

arXiv: 1902.05047
Digital Object Identifier: 10.1214/21-EJP654

Primary: 47D08, 47H40, 60J55


Vol.26 • 2021
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